Next: Multiphase mixtures Up: Special geometries Previous: Hedgehogs, Spirals, etc.

## Self-repeating structures

Here we describe an elegant yet unexpected approach to the calculation of polycrystal structures suggested in [#!Nesi:1991:PCM!#].

Consider an anisotropic material . Let us laminate this material with some anisotropic material in a laminate with a normal and a volume fraction of . Denote the resulting composite by . Suppose that it is possible to choose the parameters , , and the unknown material so that

 (23)

where is the tensor of rotation through an angle. The formula 23 states that lamination of the given anisotropic material with a material with properties leads to the composite with the properties equal to rotated on some angle.

If the relation 23 is satisfied, then is a polycrystal assembled of variously oriented fragments of .

To show this, we again use the process with an infinite number of steps. At the first step, we obtain the composite 23. At the second step, we laminate the composite with the rotated material in the same way as in the first step. Obviously, the result of this lamination is , i.e., the material rotated on the double angle. This step can be repeated infinitely many times; still, the resulting material remains equal to the rotated material .

Let us calculate the volume fractions of materials in a composite obtained at the th step. The material is used only on the first step of the procedure. Its volume fraction is equal to in the composite obtained after the first step. After the second step this fraction becomes , because the material has been added to the mixture in the volume fraction . Similarly, after the th iteration the fraction of the initial seed'' is equal to . When increases, the volume fraction of the seed'' becomes arbitrarily small, and almost all the volume is occupied with differently oriented fragments of material . In the limit , the described composite becomes a polycrystal of .

To find this polycrystal, we solve equation 23 for the unknown tensor . The parameter and the rotation tensor can be arbitrarily assigned. Any solution represents a polycrystal of . The set of solutions represents a class of polycrystals of that can be obtained by the described method.

Here, an animated graph will show the process of formation of a repeated structure

Next: Multiphase mixtures Up: Special geometries Previous: Hedgehogs, Spirals, etc.
Andre Cherkaev
2001-07-31